Finite Group Databasebeta

← #29730168

Subgroups — #29730168

Order 15616.

914 conjugacy classes of subgroups.

The subgroup lattice has 714 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840062C211yesmaximal, characteristic, complemented
2C2×C8×D244C211yesmaximal, characteristic, complemented
3#21450980C211yesmaximal, characteristic, complemented
4#21450716C211yesmaximal, characteristic
5#21447258C211yesmaximal, characteristic, complemented
6#21447251C211yesmaximal, characteristic, complemented
7#21447006C211yesmaximal, characteristic, complemented
8#21446999C211yesmaximal, characteristic, complemented
9#21446784C211yesmaximal, characteristic, complemented
10#21446587C211yesmaximal, characteristic, complemented
11#21446244C211yesmaximal, characteristic, complemented
12#21446244C211yesmaximal, characteristic, complemented
13#21445847C211yesmaximal, characteristic, complemented
14#21445843C211yesmaximal, characteristic, complemented
15C4×#102×C61C211yesmaximal, characteristic, complemented
16C22×C8×D61C2211yescharacteristic
17C22×C8×D61C2211yescharacteristic
18C2×#14519769C2211yescharacteristic
19C2×#14519748C2211yescharacteristic
20C2×#14519748C2211yescharacteristic
21C2×C4×D244C2211yescharacteristic
22#16840062—24no—
23C8×D244—24no—
24C2×#14519710C2211yescharacteristic
25C2×#14519710C2211yescharacteristic
26C2×#14519708C2211yescharacteristic
27C2×#14519708C2211yescharacteristic
28C2×C4⋊#92418C2211yescharacteristic
29C2×#14519698C2211yescharacteristic
30C2×#14519698C2211yescharacteristic
31C2×#14519689C2211yescharacteristic
32C2×#14519689C2211yescharacteristic
33#16839702—21no—
34#16839687—21no—
35#16839680—21no—
36#16839671—21no—
37#16839653—21no—
38#16839653—21no—
39#16839641—21no—
40#16839641—21no—
41#16839628C2211yescharacteristic
42#16839619C2211yescharacteristic
43#16839619C2211yescharacteristic
44#16839619C2211yescharacteristic
45#16839619C2211yescharacteristic
46#16839619C2211yescharacteristic
47#16839619C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839525C2211yescharacteristic
51#16839525C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839524C2211yescharacteristic
54C2×C4×D4×C61C2211yescharacteristic
55C2×#102×C61C2211yescharacteristic
56C2×#102×C61C2211yescharacteristic
57C2×C4×C8×C61C2211yescharacteristic
58C4×C4⋊C4×C61C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C22×D244D411yescharacteristic
62C22×C22⋊D61—21no—
63C22×C22⋊D61—21no—
64C22×C4×D61C2311yescharacteristic
65C22×C4×D61C2311yescharacteristic
66C22×#92418C2311yescharacteristic
67C22×#92418C2311yescharacteristic
68C2×C8×D61—22no—
69C2×C8×D61—22no—
70C2×C8×D61—22no—
71C2×C8×D61—22no—
72C2×C8×D61—41no—
73C2×C8×D61—41no—
74#14519769—22no—
75#14519769—22no—
76#14519748—22no—
77#14519748—22no—
78#14519748—41no—
79#14519748—41no—
80C4×D244—22no—
81C4×D244—22no—
82C2×C4⋊Dic61D411yescharacteristic
83C2×C4⋊Dic61C2311yescharacteristic
84C2×C4⋊Dic61C2311yescharacteristic
85C2×C4⋊Dic61C2311yescharacteristic
86C2×C4⋊Dic61C2311yescharacteristic
87C2×#11758196—21no—
88C2×#11758196—21no—
89C2×C22⋊Dic61—21no—
90C2×C22⋊Dic61—21no—
91C2×#11758198—21no—
92C2×#11758198—21no—
93C2×#11758198—21no—
94C2×#11758198—21no—
95C2×#11758198—21no—
96C2×#11758198—21no—
97C2×#11758198D411yescharacteristic
98C2×#11758198D411yescharacteristic
99#14519710—22no—
100#14519710—41no—
101#14519708—22no—
102#14519708—41no—
103C4⋊#92418D414yescomplemented
104#14519698—22no—
105#14519698—22no—
106#14519689—22no—
107#14519689—22no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112#14519684—21no—
113#14519684—21no—
114#14519684—21no—
115#14519684—21no—
116C22×D4×C61C2311yescharacteristic
117C23×C4×C61—21no—
118C22×C8×C61C2311yescharacteristic
119C22×C8×C61C2311yescharacteristic
120C4×D4×C61—22no—
121C4×D4×C61—22no—
122C2×C4⋊C4×C61C2311yescharacteristic
123C2×C4⋊C4×C61C2311yescharacteristic
124C2×C4⋊C4×C61C2311yescharacteristic
125C2×C22⋊C4×C61—21no—
126C2×C42×C61—21no—
127C2×C42×C61C2311yescharacteristic
128#102×C61—22no—
129#102×C61—22no—
130C4×C8×C61—22no—
131#95×C61—21no—
132C24×C61—21no—
133C23×D61C2×D411yescharacteristic
134C23×D61C2×D411yescharacteristic
135C2×D4×C61—21no—
136C2×D4×C61—21no—
137C2×D4×C61—21no—
138C2×D4×C61—21no—
139C2×D4×C61—21no—
140C2×D4×C61#4112yes—
141C22×C4×C61—21no—
142C22×C4×C61—21no—
143C22×C4×C61—21no—
144C22×C4×C61—42no—
145C22×C4×C61—81no—
146C22×C4×C61—81no—
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2411yescharacteristic
150C2×C22⋊D61—42no—
151C2×C22⋊D61—42no—
152C2×C22⋊D61—42no—
153C2×C22⋊D61—42no—
154C2×C22⋊D61—81no—
155C2×C22⋊D61—81no—
156C22×Dic61—21no—
157C22×Dic61—21no—
158C22×Dic61—21no—
159C22×Dic61—21no—
160C22×Dic61C2×D411yescharacteristic
161C22×Dic61C2×D411yescharacteristic
162C2×D244—22no—
163C2×D244—22no—
164C2×D244—22no—
165C2×C4×D61—21no—
166C2×C4×D61—21no—
167C2×C4×D61—21no—
168C2×C4×D61—21no—
169C2×C4×D61—21no—
170C2×C4×D61—21no—
171C2×C4×D61—21no—
172C2×C4×D61—21no—
173C2×C4×D61—21no—
174C2×C4×D61—21no—
175C2×C4×D61—21no—
176C2×C4×D61—21no—
177C2×C8×C61—21no—
178C2×C8×C61—21no—
179C2×C8×C61—21no—
180C2×C8×C61—21no—
181C2×C8×C61—21no—
182C2×C8×C61#4112yes—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—21no—
186C4⋊C4×C61—21no—
187C4⋊C4×C61—21no—
188C4⋊C4×C61—21no—
189C22⋊C4×C61—42no—
190C42×C61—81no—
191C42×C61C2×D412yes—
192C42×C61#4112yes—
193C22⋊Dic61—42no—
194C22⋊Dic61—42no—
195#11758198—22no—
196#11758198—22no—
197#11758198—42no—
198#11758198—42no—
199#11758198—81no—
200#11758198—81no—
201#11758198—81no—
202#11758198—81no—
203C4⋊Dic61—22no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61—21no—
208C4⋊Dic61—21no—
209C4⋊Dic61—21no—
210C4⋊Dic61—21no—
211C4⋊Dic61—21no—
212#11758196—42no—
213#11758196—42no—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418C2×D412yes—
219C2×#92418C2×D412yes—
220C8×D61—44no—
221C8×D61—44no—
222C23×C61—22no—
223C23×C61—41no—
224C23×C61—41no—
225C23×C61—41no—
226C23×C61—41no—
227C23×C61—41no—
228C23×C61—41no—
229C23×C61C22×D411yescharacteristic
230C22×D61—21no—
231C22×D61—21no—
232C22×D61—21no—
233C22×D61—21no—
234C22×D61—21no—
235C22×D61—21no—
236C22×D61—21no—
237C22×D61—21no—
238C22×D61—21no—
239C22×D61—21no—
240C22×D61—21no—
241C22×D61—21no—
242C22×D61#13512yes—
243C22×D61#13512yes—
244D4×C61—22no—
245D4×C61—22no—
246D4×C61—41no—
247D4×C61—41no—
248C2×C4×C61—21no—
249C2×C4×C61—21no—
250C2×C4×C61—21no—
251C2×C4×C61—21no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61—41no—
256C2×C4×C61—41no—
257C2×C4×C61—41no—
258C2×C4×C61—41no—
259C2×C4×C61—41no—
260C2×C4×C61—41no—
261C2×C4×C61—84no—
262C2×C4×C61#12112yes—
263C2×C4×C61#12112yes—
264C2×C4×C61C22×D411yescharacteristic
265C2×C4×C61C2×#4111yescharacteristic
266C2×C4×C61C2×#4111yescharacteristic
267C2×C4×C61C2×#4111yescharacteristic
268C2×C4×C61H(5,𝔽2)11yescharacteristic
269C2×C4×C612−1+411yescharacteristic
270C22⋊D61—84no—
271C22⋊D61—84no—
272C2×Dic61—21no—
273C2×Dic61—21no—
274C2×Dic61—21no—
275C2×Dic61—21no—
276C2×Dic61—21no—
277C2×Dic61—21no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61—41no—
282C2×Dic61—41no—
283C2×Dic61—41no—
284C2×Dic61—41no—
285C2×Dic61—41no—
286C2×Dic61—41no—
287C2×Dic61—41no—
288C2×Dic61—41no—
289C2×Dic61—41no—
290D244—28no—
291C4×D61—41no—
292C4×D61—41no—
293C4×D61—41no—
294C4×D61—41no—
295C4×D61—41no—
296C4×D61—41no—
297C4×D61—41no—
298C4×D61—41no—
299C8×C61—22no—
300C8×C61—41no—
301#92418—22no—
302#92418—22no—
303#25342—611nomaximal, Sylow
304C22×C61—42no—
305C22×C61—42no—
306C22×C61—42no—
307C22×C61—81no—
308C22×C61—81no—
309C22×C61—81no—
310C22×C61—81no—
311C22×C61C2×#12111yescharacteristic
312C22×C61#54611yescharacteristic
313C22×C61#54711yescharacteristic
314C22×C61C2×#13511yescharacteristic
315C22×C61C2×#13511yescharacteristic
316C22×C61#57611yescharacteristic
317C22×C61#57711yescharacteristic
318C2×D61—22no—
319C2×D61—22no—
320C2×D61—22no—
321C2×D61—22no—
322C2×D61—22no—
323C2×D61—22no—
324C2×D61—41no—
325C2×D61—41no—
326C2×D61—41no—
327C2×D61—41no—
328C2×D61—41no—
329C2×D61—41no—
330C2×D61—41no—
331C2×D61—41no—
332C4×C61—22no—
333C4×C61—22no—
334C4×C61—81no—
335C4×C61—81no—
336C4×C61—81no—
337C4×C61#52911yescharacteristic
338C4×C61#54211yescharacteristic
339C4×C61#54611yescharacteristic
340C4×C61#54811yescharacteristic
341Dic61—41no—
342Dic61—41no—
343Dic61—81no—
344Dic61—81no—
345Dic61—81no—
346Dic61—81no—
347C2×#478—611no—
348C2×C8×D4—611no—
349#2562—611no—
350#2535—611no—
351#2077—611no—
352#2076—611no—
353#2034—611no—
354#2033—611no—
355#1991—611no—
356#1949—611no—
357#1893—611no—
358#1893—611no—
359#1808—611no—
360#1807—611no—
361C4×#102D6111yescharacteristic, complemented
362C2×C61—82no—
363C2×C61#242711yescharacteristic
364C2×C61#305011yescharacteristic
365C2×C61#306211yescharacteristic
366C2×C61#329611yescharacteristic
367C2×C61#329711yescharacteristic
368C2×C61#333711yescharacteristic
369C2×C61#333911yescharacteristic
370D61—42no—
371D61—42no—
372C23×C8—611no—
373C23×C8—611no—
374C2×#125—611no—
375C2×#121—611no—
376C2×#121—611no—
377C2×C4×D4—611no—
378C2×C4×D4C2×D6111yescharacteristic
379#478—1224no—
380C8×D4—1224no—
381C2×#107—611no—
382C2×#107—611no—
383C2×#106—611no—
384C2×#106—611no—
385C2×C4⋊C8—611no—
386C2×#102—611no—
387C2×#102—611no—
388C2×#102C2×D6111yescharacteristic
389C2×#102C2×D6111yescharacteristic
390C2×C22⋊C8—611no—
391C2×C22⋊C8—611no—
392C2×C4×C8C2×D6111yescharacteristic
393#400—1221no—
394#397—1221no—
395#394—1221no—
396#392—1221no—
397#388—1221no—
398#388—1221no—
399#386—1221no—
400#386—1221no—
401#382—611no—
402#380—611no—
403#380—611no—
404#380—611no—
405#380—611no—
406#380—611no—
407#380—611no—
408C4×C4⋊C4C2×D6111yescharacteristic
409#340—611no—
410#340—611no—
411#340—611no—
412#340—611no—
413#340—611no—
414#340—611no—
415#340C2×D6111yescharacteristic
416#340C2×D6111yescharacteristic
417C61#2534211yescharacteristic, complemented, Sylow
418C22×D4—611no—
419C22×D4—1221no—
420C22×D4—1221no—
421C22×D4C22×D6111yescharacteristic
422C23×C4—21no—
423C23×C4—611no—
424C23×C4—611no—
425C22×C8—611no—
426C22×C8—611no—
427C22×C8—1222no—
428C22×C8—1222no—
429C22×C8—1222no—
430C22×C8—1222no—
431C22×C8—2441no—
432C22×C8—2441no—
433C22×C8C22×D6111yescharacteristic
434C22×C8C22×D6111yescharacteristic
435#125—1222no—
436#125—1222no—
437#121—1222no—
438#121—1222no—
439#121—2441no—
440#121—2441no—
441C4×D4—22no—
442C4×D4—22no—
443C4×D4—1222no—
444C4×D4—1222no—
445C2×C4⋊C4—611no—
446C2×C4⋊C4—611no—
447C2×C4⋊C4—611no—
448C2×C4⋊C4—611no—
449C2×C4⋊C4—611no—
450C2×C4⋊C4—1221no—
451C2×C4⋊C4—1221no—
452C2×C4⋊C4C22×D6111yescharacteristic
453C2×C4⋊C4C22×D6111yescharacteristic
454C2×C4⋊C4C22×D6111yescharacteristic
455C2×C22⋊C4—21no—
456C2×C22⋊C4—611no—
457C2×C22⋊C4—611no—
458C2×C22⋊C4—1221no—
459C2×C22⋊C4—1221no—
460C2×C22⋊C4—1221no—
461C2×C22⋊C4—1221no—
462C2×C22⋊C4—1221no—
463C2×C22⋊C4—1221no—
464C2×C22⋊C4—1221no—
465C2×C22⋊C4—1221no—
466C2×C42—21no—
467C2×C42C22×D6111yescharacteristic
468#107—1222no—
469#107—2441no—
470#106—1222no—
471#106—2441no—
472C4⋊C8—614no—
473#102—22no—
474#102—22no—
475#102—1222no—
476#102—1222no—
477C22⋊C8—1222no—
478C22⋊C8—1222no—
479C4×C8—22no—
480#95—21no—
481#95—1221no—
482#95—1221no—
483#95—1221no—
484#95—1221no—
485#95—1221no—
486#95—1221no—
487#95—1221no—
488#95—1221no—
489C24—21no—
490C24—611no—
491C24—611no—
492C2×D4—21no—
493C2×D4—21no—
494C2×D4—21no—
495C2×D4—21no—
496C2×D4—21no—
497C2×D4—1222no—
498C2×D4—1222no—
499C2×D4—1222no—
500C2×D4—4881no—
501C2×D4—4881no—
502C2×D4—2442no—
503C2×D4—2442no—
504C2×D4—2442no—
505C2×D4—2442no—
506C2×D4#1175822212yes—
507C22×C4—21no—
508C22×C4—21no—
509C22×C4—21no—
510C22×C4—42no—
511C22×C4—81no—
512C22×C4—81no—
513C22×C4—611no—
514C22×C4—611no—
515C22×C4—1221no—
516C22×C4—1221no—
517C22×C4—1221no—
518C22×C4—1221no—
519C22×C4—1221no—
520C22×C4—1221no—
521C22×C4—1221no—
522C22×C4—1221no—
523C22×C4—1221no—
524C22×C4—1221no—
525C22×C4—1221no—
526C22×C4—1221no—
527C22×C4—1221no—
528C22×C4—1221no—
529C22×C4—1221no—
530C22×C4—1221no—
531C22×C4D4×D6111yescharacteristic
532C22×C4D4×D6111yescharacteristic
533C22×C4C23×D6111yescharacteristic
534C2×C8—21no—
535C2×C8—21no—
536C2×C8—21no—
537C2×C8—21no—
538C2×C8—21no—
539C2×C8—612no—
540C2×C8—612no—
541C2×C8—1221no—
542C2×C8—1221no—
543C2×C8—1221no—
544C2×C8—1221no—
545C2×C8—2444no—
546C2×C8—2444no—
547C2×C8#1175822212yes—
548C4⋊C4—21no—
549C4⋊C4—21no—
550C4⋊C4—21no—
551C4⋊C4—21no—
552C4⋊C4—21no—
553C4⋊C4—21no—
554C4⋊C4—1222no—
555C4⋊C4—1221no—
556C4⋊C4—1221no—
557C4⋊C4—1221no—
558C4⋊C4—1221no—
559C4⋊C4—1221no—
560C4⋊C4—1221no—
561C4⋊C4—1221no—
562C4⋊C4—1221no—
563C4⋊C4—2442no—
564C4⋊C4—2442no—
565C22⋊C4—42no—
566C22⋊C4—1222no—
567C22⋊C4—1222no—
568C22⋊C4—4881no—
569C22⋊C4—4881no—
570C22⋊C4—4881no—
571C22⋊C4—4881no—
572C22⋊C4—2442no—
573C22⋊C4—2442no—
574C22⋊C4—2442no—
575C22⋊C4—2442no—
576C42—81no—
577C42#1175822212yes—
578C42D4×D6112yes—
579C23—22no—
580C23—41no—
581C23—41no—
582C23—41no—
583C23—41no—
584C23—41no—
585C23—41no—
586C23—612no—
587C23—612no—
588C23—1221no—
589C23—1221no—
590C23—1221no—
591C23—1221no—
592C23—1221no—
593C23—1221no—
594C23—1221no—
595C23—1221no—
596C23—1221no—
597C23—1221no—
598C23—1221no—
599C23—1221no—
600C23C2×D4×D6111yescharacteristic
601D4—22no—
602D4—22no—
603D4—41no—
604D4—41no—
605D4—1228no—
606D4—4884no—
607D4—4884no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—84no—
622C2×C4—1221no—
623C2×C4—1221no—
624C2×C4—1221no—
625C2×C4—1221no—
626C2×C4—1221no—
627C2×C4—1221no—
628C2×C4—2441no—
629C2×C4—2441no—
630C2×C4—2441no—
631C2×C4—2441no—
632C2×C4—2441no—
633C2×C4—2441no—
634C2×C4—2441no—
635C2×C4—2441no—
636C2×C4—2441no—
637C2×C4—2441no—
638C2×C4—2441no—
639C2×C4—2441no—
640C2×C4—2441no—
641C2×C4—2441no—
642C2×C4—2441no—
643C2×C4—2441no—
644C2×C4—2441no—
645C2×C4—2441no—
646C2×C4—2441no—
647C2×C4—2441no—
648C2×C4#1451975012yes—
649C2×C4#1451975012yes—
650C2×C4C2×D4×D6111yescharacteristic
651C2×C4C2×#1175822211yescharacteristic
652C2×C4C2×#1175822211yescharacteristic
653C2×C4C2×#1175822211yescharacteristic
654C2×C4#1451982311yescharacteristic
655C2×C4#1451982411yescharacteristic
656C8—22no—
657C8—41no—
658C8—1222no—
659C8—1222no—
660C22—42no—
661C22—42no—
662C22—42no—
663C22—81no—
664C22—81no—
665C22—81no—
666C22—81no—
667C22—1222no—
668C22—1222no—
669C22—1222no—
670C22—1222no—
671C22—1222no—
672C22—1222no—
673C22—2441no—
674C22—2441no—
675C22—2441no—
676C22—2441no—
677C22—2441no—
678C22—2441no—
679C22—2441no—
680C22—2441no—
681C22C2×#1451975011yescharacteristic
682C22#1684040811yescharacteristic
683C22#1684041711yescharacteristic
684C22#135×D6111yescharacteristic
685C22#135×D6111yescharacteristic
686C22#1684056311yescharacteristic
687C22#1684056811yescharacteristic
688C4—22no—
689C4—22no—
690C4—81no—
691C4—81no—
692C4—81no—
693C4—4881no—
694C4—4881no—
695C4—4881no—
696C4—4881no—
697C4—2441no—
698C4—2441no—
699C4#1684029611yescharacteristic
700C4#1684038311yescharacteristic
701C4#1684040811yescharacteristic
702C4#1684042111yescharacteristic
703C2—82no—
704C2—2442no—
705C2—2442no—
706C2#2144939611yescharacteristic
707C2#2145539911yescharacteristic
708C2#2145549711yescharacteristic
709C2#2145765911yescharacteristic
710C2#2145767411yescharacteristic
711C2#2145820311yescharacteristic
712C2#2145822311yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC4×#102×C61C2
socleC23×C61C22×D4
prime radical#21450716C2
hypercenterC4×#102D61
hypocenterC61#25342
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#102

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#16840062C2semidirect
2. C2×C8×D244C2semidirect
3. #21450980C2semidirect
5. #21447258C2semidirect
6. #21447251C2semidirect
7. #21447006C2semidirect
8. #21446999C2semidirect
9. #21446784C2semidirect
10. #21446587C2semidirect
11. #21446244C2semidirect
12. #21446244C2semidirect
13. #21445847C2semidirect
14. #21445843C2semidirect
15. C4×#102×C61C2semidirect
103. C4⋊#92418D4semidirect
361. C4×#102D61semidirect
417. C61#25342semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168400622
C2×C8×D2442
#214509802
#214507162
#214472582
#214472512
#214470062
#214469992
#214467842
#214465872
#214462442
#214458472
#214458432
C4×#102×C612
#2534261